Engine cooling taken care of by the prop?
Take a look at the shape of the prop blade between the hub and the edge of the cowl...say the first 15- 20+cm...it is round and has no aerodynamic properties.
The only air travelling through the cowl when the aircraft is stationary and at higher than idle power settings is that which is sucked through by the venturi effect of air flowing around the cowl and passed the cowl flaps, if fitted.
At idle/taxi power settings with the aircraft moving 'at a brisk walking pace' there may be little or no cooling airflow through the cowl.
Yes parking into any significant wind will help put more air through the cowl...although I suspect the strength of wind required to actually make a difference might be stronger than you'd want to be flying in in the first place.
As to wind effects on controls..ailerons excepted...think about how strong the wind flowing around the airframe from prop blast is at the various power settings is. Even at idle it would be 10kts probably. At typical 'mag check' power settings it would be closer to 30-40 kts. I doubt being parked in a 10 kt tailwind would make much difference.
The better question to ask is why pilots park in 'run up bays' to do their engine/prop checks in the first place.
I would suggest it is a habit that is formed in the first hours of learning to fly because students are deemed to be incapable of taxiing and doing runups concurrently. That is probably a reasonable premise but at some point in the training perhaps it needs reviewing.
It has been over 25 years since I have parked into wind, whether in a runup bay or not, and done a classic runup. The vast preponderance of experienced/commercial pilots do them on the run.
This is viewed as less than 'professional' by many instructors but when you think about engine cooling on the ground perhaps getting the aircraft off the ground in reasonably short order bares some consideration.